Informative cluster sizes for subcluster-level covariates and weighted generalized estimating equations

Williamson, Datta, and Satten's (2003, Biometrics 59, 36-42) cluster-weighted generalized estimating equations (CWGEEs) are effective in adjusting for bias due to informative cluster sizes for cluster-level covariates. We show that CWGEE may not perform well, however, for covariates that can take different values within a cluster if the numbers of observations at each covariate level are informative. On the other hand, inverse probability of treatment weighting accounts for informative treatment propensity but not for informative cluster size. Motivated by evaluating the effect of a binary exposure in presence of such types of informativeness, we propose several weighted GEE estimators, with weights related to the size of a cluster as well as the distribution of the binary exposure within the cluster. Choice of the weights depends on the population of interest and the nature of the exposure. Through simulation studies, we demonstrate the superior performance of the new estimators compared to existing estimators such as from GEE, CWGEE, and inverse probability of treatment-weighted GEE. We demonstrate the use of our method using an example examining covariate effects on the risk of dental caries among small children.